As others have said, there's no requirement for starting the Crystal Tower series other than completing the ARR main story (not patch content). The quest for Crystal Tower starts in Mor Dhona and is called "The Legacy of Allag". If you don't see it, you may have already picked it up, so check your journal to see which quests you have active.
At level 50, it can seem overwhelming with all the blue quests that pop up all of a sudden. Most of them just unlock optional dungeons or extreme trials that you can do at your leisure (note that extreme trials are done in the party finder, not with duty finder). The important things to finish for MSQ are the Crystal Tower series and the three "hard" mode primal trials for Ifrit, Garuda, and Titan. All of them can be done in duty finder, although you should think about data center traveling to a world on the Aether data center to queue up for them, so you can avoid lengthy queues (hopefully queues on the Dynamis data center will improve with the expansion, but for now the problem is that most people on Dynamis are going to Aether to queue for their roulettes).
ETA: Marilith is the world you're on, Dynamis is the data center. To travel you need to be logged out and right-click on your character name and select 'Visit Another Data Center'. From there, click on 'Select World' and it'll prompt you to select the data center and world that you want to travel to.
ETA 2: I'm not sure how late you're usually on, but if it's in the evenings and not past midnight, I might be able to help occasionally. Let me know if you want my discord info.
